Properties

Label 173.17
Modulus $173$
Conductor $173$
Order $172$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(173, base_ring=CyclotomicField(172))
 
M = H._module
 
chi = DirichletCharacter(H, M([73]))
 
pari: [g,chi] = znchar(Mod(17,173))
 

Basic properties

Modulus: \(173\)
Conductor: \(173\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(172\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 173.f

\(\chi_{173}(2,\cdot)\) \(\chi_{173}(3,\cdot)\) \(\chi_{173}(5,\cdot)\) \(\chi_{173}(7,\cdot)\) \(\chi_{173}(8,\cdot)\) \(\chi_{173}(11,\cdot)\) \(\chi_{173}(12,\cdot)\) \(\chi_{173}(17,\cdot)\) \(\chi_{173}(18,\cdot)\) \(\chi_{173}(19,\cdot)\) \(\chi_{173}(20,\cdot)\) \(\chi_{173}(26,\cdot)\) \(\chi_{173}(27,\cdot)\) \(\chi_{173}(28,\cdot)\) \(\chi_{173}(30,\cdot)\) \(\chi_{173}(32,\cdot)\) \(\chi_{173}(39,\cdot)\) \(\chi_{173}(42,\cdot)\) \(\chi_{173}(44,\cdot)\) \(\chi_{173}(45,\cdot)\) \(\chi_{173}(46,\cdot)\) \(\chi_{173}(48,\cdot)\) \(\chi_{173}(50,\cdot)\) \(\chi_{173}(53,\cdot)\) \(\chi_{173}(58,\cdot)\) \(\chi_{173}(59,\cdot)\) \(\chi_{173}(61,\cdot)\) \(\chi_{173}(62,\cdot)\) \(\chi_{173}(63,\cdot)\) \(\chi_{173}(65,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{172})$
Fixed field: Number field defined by a degree 172 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{73}{172}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 173 }(17, a) \) \(-1\)\(1\)\(e\left(\frac{73}{172}\right)\)\(e\left(\frac{79}{172}\right)\)\(e\left(\frac{73}{86}\right)\)\(e\left(\frac{95}{172}\right)\)\(e\left(\frac{38}{43}\right)\)\(e\left(\frac{55}{172}\right)\)\(e\left(\frac{47}{172}\right)\)\(e\left(\frac{79}{86}\right)\)\(e\left(\frac{42}{43}\right)\)\(e\left(\frac{131}{172}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 173 }(17,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 173 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 173 }(17,·),\chi_{ 173 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 173 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2